I recently started SF and have a question about when to move up to the next level. Can you only move to next level if both Day2 and Day3 are under 5 mins? Or, is it only Day3 which determines whether you can move up to next level?

Clicking knees:
I only have one piece of advice, and that is to align your feet under the path your knees travel as you lower into the squat. If you are already doing that, then I am afraid I have nothing else to offer.

I did Day3 of Level 2 today @ 3min 22sec. It was super tough - felt like throwing up after.

I cheated a lot on my pullups and I need some advice on whether I deserve to go up a level or remain on level 2:

For days 1 & 2 I do full and proper pullups. For Day 3, I needed to do 13 pullups, and I did them in this pattern: 3,2,1,1,1,1,1,1,1,1. The first 3,2,1 sets were strict pullup form. But the remaining 1's were really bad form: from standing position I jumped up to the pullup bar and used the momentum to reach my chin to bar - in all honesty they were probably 'three-quarter' pullups instead of full pullups - they were still very hard: I had to rest after each rep and nearly reached failure. I considered kipping, but as my doorway pullup bar is not that sturdy I'm afraid of breaking it (and myself).

Given that I did so many 'three-quarter' pullups, should I redo Level 2? And if so, how should I do Level 3 pullups? Do 3,2,1 with proper form, and the remaining sets use the 'three-quarter' pullups, but 3 reps =1 regular rep? (however, this 3=1 is very daunting as like I said above even these three-quarter pullups are super difficult for me).

Getting to the first pull-up is hard. Once you can do 1 good pull-up, you've got the tool you need to move ahead, without cheating.
There is nothing wrong with doing the Day 3 pull-ups one by one, but you need to rest enough between reps to allow you to maintain decent form. Obviously this will affect your time, but who knows, it might still be sub-five-minutes.